In this paper, a fundamentally new method, based on the definition, is introduced for numerical computation of Eigenvalues, generalized Eigenvalues and quadratic Eigenvalues of matrices. Some examples are provided to show the accuracy and reliability of the proposed method. It is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to the desired Eigenvalues, generalized Eigenvalues and quadratic Eigenvalues of matrices. These examples show an interesting phenomenon in the procedure: The diagonal matrix that converges to Eigenvalues gives them in decreasing order in the sense of absolute value. Appendices A to C provide Matlab codes that implement the proposed algorithms. They show that the proposed algorithms are very easy to program.

The aim of this paper is to determine an upper bound for the number of non-co-spectral permutation graphs in terms of automorphism group of a graph G. As a corollary, we determine the Eigenvalues of all permutation graphs P (Cn), where  2 Aut(Cn).

In this paper, we study the Eigenvalues of real tridiagonal 3-Toeplitz matrices of different order. When the order of a tridiagonal 3-Toeplitz matrix is n = 3k+ 2, the Eigenvalues were found explicitly. Here, we consider the distribution of Eigenvalues for a tridiagonal 3-Toeplitz matrix of orders n = 3k and n = 3k + 1. We explain our method by finding roots of a combination of Chebyshev polynomials of the second kind. This distribution solves the eigenproblem for integer powers of such matrices.

The sharp upper bounds and the sharp lower bounds of the largest Eigenvalues l1, the least eigenvalue ln, the second largest eigenvalue l2, the spread and the separator among all firefly graphs on n vertices are determined.